Differentiation of sinx | differentiation of sinx by first principle
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In this video u will learn Derivative of Sin(x) using first principle method @Epselon
The derivative of sin x is denoted by d/dx (sin x) = cos x. The other way to represent the sine function is (sin x)' = cos x.
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Derivative of Sin x Proof by Chain Rule :
Using one of the trigonometric formulas, we can write sin x as, sin x = cos (π/2 - x). Using this let us find the derivative of y = sin x (or) cos (π/2 - x). = cos x (By one of the trigonometric formulas). Thus, we have derived the formula of derivative of sin x by chain rule.
Using chain rule,
y' = - sin(π/2 - x) · d/dx (π/2 - x) (as the derivative of cos x is - sin x)
= - sin(π/2 - x) · (-1)
= sin(π/2 - x)
= cos x (By one of the trigonometric formulas).
Thus, we have derived the formula of derivative of sin x by chain rule.
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Derivative of Sin x Proof by Quotient Rule
The quotient rule says d/dx (u/v) = (v u' - u v') / v2. So to find the differentiation of sin x using the quotient rule, we have to write sin x as a fraction. We know that sin is the reciprocal of the cosecant function (csc). i.e., y = sin x = 1/(csc x). Then by chain rule,
y' = [csc x · d/dx(1) - 1 · d/dx(csc x)] / csc2x
= [csc x (0) - 1 (-csc x cot x)] / csc2x (as the derivative of csc x is -csc x cot x]
= (cot x) / (csc x)
= [(cos x)/(sin x)] / [1/sin x]
= cos x
Therefore, the derivative of sin is cos x and is proved by using the quotient rule.
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Derivative of sinx by first principle :
Now, we have to find the derivative of sin (x+1), using the 1st principle. f'(x) = cos (x+1). Hence, the derivative of sin (x+1), with respect to x is cos (x+1).
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